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Proofgold Proof

pf
Let x0 of type ιο be given.
Let x1 of type ιιι be given.
Assume H0: ∀ x2 x3 . x0 x2x0 x3x0 (x1 x2 x3).
Assume H1: ∀ x2 x3 x4 . x0 x2x0 x3x0 x4x1 x2 (x1 x3 x4) = x1 x3 (x1 x2 x4).
Assume H2: ∀ x2 x3 . x0 x2x0 x3x1 x2 x3 = x1 x3 x2.
Let x2 of type ι be given.
Let x3 of type ι be given.
Let x4 of type ι be given.
Let x5 of type ι be given.
Let x6 of type ι be given.
Let x7 of type ι be given.
Let x8 of type ι be given.
Assume H3: x0 x2.
Assume H4: x0 x3.
Assume H5: x0 x4.
Assume H6: x0 x5.
Assume H7: x0 x6.
Assume H8: x0 x7.
Assume H9: x0 x8.
Apply H1 with x6, x7, x8, λ x9 x10 . x1 x2 (x1 x3 (x1 x4 (x1 x5 x10))) = x1 x8 (x1 x2 (x1 x7 (x1 x5 (x1 x3 (x1 x4 x6))))) leaving 4 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
Apply H2 with x6, x8, λ x9 x10 . x1 x2 (x1 x3 (x1 x4 (x1 x5 (x1 x7 x10)))) = x1 x8 (x1 x2 (x1 x7 (x1 x5 (x1 x3 (x1 x4 x6))))) leaving 3 subgoals.
The subproof is completed by applying H7.
The subproof is completed by applying H9.
Apply unknownprop_b835917d9e2c641c52f0a836abeb41c4c8a4498d6b8d8c6014d0e529d788822d with x0, x1, x2, x3, x4, x5, x7, x8, x6 leaving 9 subgoals.
The subproof is completed by applying H0.
The subproof is completed by applying H1.
The subproof is completed by applying H3.
The subproof is completed by applying H4.
The subproof is completed by applying H5.
The subproof is completed by applying H6.
The subproof is completed by applying H8.
The subproof is completed by applying H9.
The subproof is completed by applying H7.